There are many papers pure and applied on this topic. I guess you are looking for applications.

You can

1. measure the changes of neural network output with respect to the perturbation of single input variables of the mean object for each class of training.

2. measure the average sensitivity of the class objects.

3. or Follow

M. Gevrey, I. Dimopoulos, and S. Lek, “Review and comparison

of methods to study the contribution of variables in artificial

neural network models,” Ecological Modelling, vol. 160, no. 3, pp.

249–264, 2003.

If you are  interested in decision making applications, then see for example:

Iranian Journal of Operations Research

Vol. 4, No. 1, 2013, pp. 14-24

There is also the more fundamental question of whether there in fact is a correct sensitivity formula at all. If the model represented by the ANN provides not only a unique solution for every given input "near" a base case, but also one that is stable enough to allow a kind of derivative to exist in the direction of parameter movement at that base case, THEN one can discuss (and perhaps also calculate) the sensitivity analysis discussed in the question. Whether that is the case depends crucially on the properties of the original model, of course. 

Hello,

You will find in the paper below a sensitivity measure based on "What If ?" simulations.

In short, take the sum over a of the expected value w.r.t. p(x_i) of the difference (f_w(x_i+a)-f_w(x_i))., f_w denotes a function, for instance a MLP with weights w.

See:

https://www.researchgate.net/publication/11348549_A_methodology_to_explain_neural_network_classification

Article A methodology to explain neural network classification

There are many papers applied on this topic one of them is the article

 Sensitivity Analysis of the Artificial Neural Network Outputs in

Friction Stir Lap Joining of Aluminum to Brass

Mohammad Hasan Shojaeefard,1 Mostafa Akbari,1 Mojtaba Tahani,1,2 and Foad Farhani3

I have formulated a dependancy matrix, which allows you to calculate the sensitivity between dependant and independant veriales in a multilayer perceptron network.Look at my publications. This desing and implementation goes back to 1992 and is applied in many complex industrial applications, that I did in the past decades.

Maybe this might help you in your circumstance.

There are many papers applied on this topic one of them is the article